Extreme Events for Fractional Brownian Motion with Drift: Theory and Numerical Validation

TL;DR

This paper develops a theoretical framework and numerical validation for the first-passage time, maximum distribution, and absorption probability of fractional Brownian motion with drift, using perturbative expansion and advanced simulation algorithms.

Contribution

It provides the first-order analytical corrections for fractional Brownian motion with drift and introduces an efficient adaptive bisection algorithm for high-precision numerical validation.

Findings

Excellent agreement between theory and simulations.

First-order corrections improve classical Brownian motion results.

Efficient algorithm enables large-scale validation with over 2.7×10^8 points.

Abstract

We study the first-passage time, the distribution of the maximum, and the absorption probability of fractional Brownian motion of Hurst parameter $H$ with both a linear and a non-linear drift. The latter appears naturally when applying non-linear variable transformations. Via a perturbative expansion in $\epsilon = H-1/2$, we give the first-order corrections to the classical result for Brownian motion analytically. Using a recently introduced adaptive bisection algorithm, which is much more efficient than the standard Davies-Harte algorithm, we test our predictions for the first-passage time on grids of effective sizes up to $N_{\rm eff}=2^{28}\approx 2.7\times 10^{8}$ points. The agreement between theory and simulations is excellent, and by far exceeds in precision what can be obtained by scaling alone.